Theory Of Surveying Data Processing With Prior Information And Its Applications In Geodesy

Abstract:The prior information of parameters is common in geodesy. This information may contain prior expectation and variance, or the equality or inequality constraints. If we deeply excavate the prior information in practice and bring such information into the observation model, the accuracy and reliability of the parameter is bound to be improved. The source of prior information in geodesy is explored and a corresponding adjustment model is established based on the form of constraints. The algorithms and accuracy assessment and the influence of the constraints to the model are analyzed. Then the constrained adjustment theory is applied to GPS data processing and deformation monitoring of control network which shows a good performance. The main content and innovative points are listed as follows:1) The source of prior information in geodesy is studied systematically and the information is divided into stochastic and functional one according to its form. The equality and inequality constrained least square(ICLS) model are emphasized in this paper. The Kuhn-Tucker condition of the optimal solution of ICLS problem is analyzed. The least distance method, penalty method, linear complementary programming(LCP), aggregate function method, elliptical constraint method and Bayesian method to ICLS problem are represented and the differences between each other are compared.2) Most of the algorithms to ICLS problems are based on optimization and far away from classic adjustment theory, so an exhaustive method which is based on equality-constrained model is proposed. Then another simple iterative method is proposed based on penalty function and the inequality constraints are treated as virtual observations. A simulation shows the feasibility and convergence of this method. Then it is applied to the deformation monitoring of the dam of a hydropower station, the inequality constrained solution is more objective and realistic. Meanwhile, the algorithms to rank deficient network with equality and inequality constraints are given.3) The influence of equality constraints to the results of adjustment is analyzed and a hypothesis procedure is designed to test if the solution has changed significantly. According to the process of adjustment and the theory of probability and statistics, the influence of inequality constraints to the results of adjustment is analyzed. Then the advantages and shortcomings of each method to ICLS problems are explored and a Monte Carlo method is utilized to analyze the accuracy of the ICLS solution.4) The regularization method to equality ill-posed problem is proposed and the statistical properties of the solution are analyzed, the regularization factor is chosen by minimizing the mean square error. Then the influence of equality constraints to the ill-posed problem is analyzed and a method to eliminate some parameters is proposed to solve the problem. A regularization method is proposed to solve the inequality constrained ill-posed problem. First a new method transforming the inequality constrained problem to a corresponding equality constrained one is proposed. Then it is proved by Bayesian method that the regularization factor is the ratio of the variance of real observations and virtual ones. At last a method to choose the regularization factor is proposed on the basis of Helmert variance component estimation.5) The relation between total least squares(TLS) and common least squares in line fitting is analyzed and two methods respectively based on singular value decomposition(SVD) and Lagrange multipliers are given to solve the TLS problem. Then a Lagrange multipliers method is proposed to solve the TLS problem with equality or inequality constraints. Finally, the Tikhonov regularization method is used to solve the ill-conditioned TLS problem.